Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $q \neq 0$. $a = \dfrac{4(4q - 7)}{6q} \div \dfrac{10(4q - 7)}{6q} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{4(4q - 7)}{6q} \times \dfrac{6q}{10(4q - 7)} $ When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ 4(4q - 7) \times 6q } { 6q \times 10(4q - 7) } $ $ a = \dfrac{24q(4q - 7)}{60q(4q - 7)} $ We can cancel the $4q - 7$ so long as $4q - 7 \neq 0$ Therefore $q \neq \dfrac{7}{4}$ $a = \dfrac{24q \cancel{(4q - 7})}{60q \cancel{(4q - 7)}} = \dfrac{24q}{60q} = \dfrac{2}{5} $